Firm Boundaries as Supply Chain Risk Management

Quinta Conferencia de Economistas Costa Rica

José Ignacio González Rojas

London School of Economics and Political Science

December 16, 2025

The Question

How Do Firms Set Their Boundaries Under Uninsurable Operational Risk? How Does This Shape Industry Structure?

Post-COVID reality

  • Shocks lasting 1+ month occur every 3.7 years (Lund et al., 2020).
  • 93% of business leaders plan to increase supply chain resilience (Lund et al., 2020).
  • But their strategies seem contradictory from the perspective of firm boundaries:
    • Intel, Samsung, Apple, Ford are specialising ⬆️
    • Hasbro, Nike, Nintendo are fragmenting ⬇️

The puzzle

  • Why do firms facing similar operational risks choose opposite organisational responses?

What’s Been Done ➕ What’s Missing

Theory of the Firm (Antràs, 2003; Bolton & Whinston, 1993; Coase, 1937; S. J. Grossman & Hart, 1986; Hart & Moore, 1990; Kikuchi et al., 2018; Williamson, 1979)

  • Transaction costs ➕ incomplete contracts ➕ ownership rights ➡️ firm boundaries
  • Focus on ex-post adaptation (bargaining, ownership)—what about ex-ante planning under uninsurable risk?

Supply Chain Organisation (Antràs & Chor, 2013; Antràs & de Gortari, 2020; Fally & Hillberry, 2018; Kikuchi et al., 2018)

  • Transaction costs ➕ span-of-control limits ➡️ firms specialise
  • Downstream firms are located in more central locations
  • How do operational risk and production technology alone create boundaries?

Supply Chain Resilience (Castro-Vincenzi, 2024; Elliott et al., 2022; Elliott & Jackson, 2024; G. M. Grossman et al., 2023, 2024; Martinez, 2024)

  • Multisourcing ➕ relationship investment ➕ spatial diversification ➡️ resilience
  • What if risk is uninsurable and firms plan ahead?
  • How do we capture the sequential nature of production and risk?

This Paper

Firm Boundaries Emerge from Technology and Risk

Key Modelling Choices

  1. Sequential production: Stages must complete in order—snakes (Baldwin & Venables, 2013)
  2. Poisson operational failures At rate \(\lambda\) per unit scope, the product is destroyed—uninsurable due to moral hazard ➕ adverse selection
  3. Organisational complexity costs \(c''(\ell) > 0\): Coordination and management costs rise at an increasing rate with scope

Joint Analysis of Firm Boundaries and Industry Structure

  • Firm boundaries: How does operational risk shape optimal vertical scope?
  • Industry structure: How many firms? Where do boundaries fall along the chain?
  • Ex-ante planning perspective: Firms choose boundaries before shocks realise, not adapt after
  • Comparative statics in risk: How do changes in operational risk affect firm boundaries choice and industry structure?

How Risk Shapes Firm Boundaries and Industry Structure

Three mechanisms create finite boundaries

  1. Restart costs: Firms expect to fail \(e^{\lambda \ell}\) times ➡️ must repurchase inputs each time
  2. Organisational constraints: \(c''(\ell) > 0\) ➡️ longer spans harder to manage
  3. Variance benefit ⭐: When firms fail early, they save on costs due to convexity ➡️ encourages longer spans

Non-monotonic response to change in frequency of failures

  • Infrequent: Variance benefit dominates ➡️ risk encourages specialisation
  • Frequent: Restart costs dominate ➡️ risk drives fragmentation
  • Very frequent: Too risky! ➡️ Sequential exit ➡️ upstream firms shut down first

Industry structure reflects risk exposure

  • Endogenous segmentation: Ex-ante identical firms specialise in different production segments
  • Downstream vulnerability: Restart costs scale with input prices \(p(s)\) ➡️ downstream firms more exposed

Intuition: The Pizza Supply Chain 🍕

FIRM 1 Raw Materials FIRM 2 Basic Processing FIRM 3 Assembly & Toppings FIRM 4 Finishing & Delivery Raw Flour Dough + Sauce + Cheese + Toppings Bake Serve 🍕 PIZZA

The Model

The Environment

Sequential stages \(s \in [0,1]\) (Becker & Murphy, 1992; Kikuchi et al., 2018)

  • Cannot produce \(b' > b\) before \(b\)
  • Homogeneous good at each stage

Organisational costs \(c(\ell)\) (Becker & Murphy, 1992; Lucas, 1978)

\[c(0) = 0, \quad c'(0) > 0, \quad c''(\ell) > 0\]

  • Constant returns to scale
  • Decreasing returns to scope (diminishing returns to management)

Operational failures \(U \sim \text{Exp}(\lambda)\) (Kremer, 1993; Sobel, 1992)

  • Complete destruction, no salvage
  • Restart ➡️ buy input from previous firm

Contestable Markets (Baumol, 1982)

  • Technology available to all firms
  • Free entry and exit
  • Zero profits ex ante (entry threat)

Firms Minimise the Number of Production Attempts \(\times\) Expected Cost per Attempt

Bellman equation

\[p(s)=\min_{0 \leq \ell \leq s} \left\{e^{\lambda \ell} \cdot (\tilde{c}(\ell) + p(b))\right\}\]

First-Order Condition

\[\underbrace{p'(b^{*})}_{\text{Marginal profit}} = \underbrace{e^{-\lambda\ell^*}c'(\ell^*)}_{\text{Marginal org cost}} + \underbrace{\lambda[p(b^{*}) + \tilde{c}(\ell^*)]}_{\text{Marginal restart cost}}\]

Comparative Statics

Why Risk Can Encourage Longer Production Runs

Variance benefit is the gap between full cost and average sunk cost

\[VB(\ell) = c(\ell) - \mathbb{E}[c(U)|U<\ell] > 0\]

Expected cost per attempt decomposes into full cost minus variance benefit

\[\begin{align*} \tilde{c}(\ell) &= e^{-\lambda\ell}c(\ell) + (1-e^{-\lambda\ell})\mathbb{E}[c(U)|U<\ell] \\ &= c(\ell) - \underbrace{(1-e^{-\lambda\ell})VB(\ell)}_{\text{Variance benefit reduction}} \end{align*}\]

How does optimal scope \(\ell^{*}\) change with the rate \(\lambda\)?

The effects of more frequent failures is ambiguous
\(\text{sign}\left(\frac{\partial\ell^{*}}{\partial \lambda}\right) = \text{sign}\left(\Psi(\ell^{*}, \lambda)\right)\)

\[\begin{align*} \Psi(\ell, \lambda) &= \frac{\partial \mathbb{P}[\text{success}]}{\partial \lambda}\left(\underbrace{c'(\ell)}_{\substack{\text{direct marginal} \\ \text{cost discount}}}+\underbrace{\lambda VB(\ell)}_{\text{variance benefit}}\right) \\ &+ \underbrace{\lambda\mathbb{P}[\text{failure}]\text{Cov}(U,c(U)|U<\ell)}_{\text{early exit option value}} \\ &- \underbrace{[p(b)+ \tilde{c}(\ell)]}_{\text{total exposure}} \end{align*}\]

Conclusion

Main Contributions

Alternative foundation

  • Technology alone (no contracts)
  • Operational risk + organizational costs

Variance benefit mechanism

  • Counterintuitive: risk can encourage specialisation
  • Depends on cost convexity, hazard rate
  • Three distinct regimes

Position matters

  • Downstream vulnerability emerges naturally
  • Input cost effect drives asymmetry
  • Explains observed restructuring patterns

Industry equilibrium

  • Higher frequency of shocks ➡️ fewer firms
  • Allocative efficient boundaries emerge naturally
  • Firms share risk exposure through prices

Thank you!

References

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